metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.14D28, C24.10D14, (C2×C28).51D4, C2.6(C28⋊7D4), (C22×C14).65D4, (C22×C4).91D14, C14.58(C4⋊D4), C22.125(C2×D28), C7⋊4(C23.11D4), C14.C42⋊15C2, C2.6(C28.17D4), C14.38(C4.4D4), C22.98(C4○D28), (C23×C14).36C22, (C22×C28).60C22, C23.370(C22×D7), C14.16(C42⋊2C2), C22.96(D4⋊2D7), (C22×C14).328C23, C2.8(C23.18D14), C14.73(C22.D4), C2.16(C22.D28), C2.14(C23.D14), (C22×Dic7).42C22, (C2×C4⋊Dic7)⋊12C2, (C2×C14).432(C2×D4), (C2×C4).30(C7⋊D4), (C2×C22⋊C4).15D7, (C14×C22⋊C4).16C2, C22.126(C2×C7⋊D4), (C2×C23.D7).15C2, (C2×C14).144(C4○D4), SmallGroup(448,487)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.14D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=c, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 692 in 170 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C7, C2×C4 [×2], C2×C4 [×17], C23, C23 [×2], C23 [×6], C14 [×3], C14 [×4], C14 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, Dic7 [×4], C28 [×3], C2×C14 [×3], C2×C14 [×4], C2×C14 [×10], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×Dic7 [×12], C2×C28 [×2], C2×C28 [×5], C22×C14, C22×C14 [×2], C22×C14 [×6], C23.11D4, C4⋊Dic7 [×2], C23.D7 [×4], C7×C22⋊C4 [×2], C22×Dic7 [×4], C22×C28 [×2], C23×C14, C14.C42, C14.C42 [×2], C2×C4⋊Dic7, C2×C23.D7 [×2], C14×C22⋊C4, C23.14D28
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D7, C2×D4 [×2], C4○D4 [×5], D14 [×3], C4⋊D4, C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], D28 [×2], C7⋊D4 [×2], C22×D7, C23.11D4, C2×D28, C4○D28, D4⋊2D7 [×4], C2×C7⋊D4, C23.D14 [×2], C22.D28 [×2], C28⋊7D4, C23.18D14, C28.17D4, C23.14D28
►On 224 points
Generators in S224
(1 15)(2 219)(3 17)(4 221)(5 19)(6 223)(7 21)(8 197)(9 23)(10 199)(11 25)(12 201)(13 27)(14 203)(16 205)(18 207)(20 209)(22 211)(24 213)(26 215)(28 217)(29 128)(30 44)(31 130)(32 46)(33 132)(34 48)(35 134)(36 50)(37 136)(38 52)(39 138)(40 54)(41 140)(42 56)(43 114)(45 116)(47 118)(49 120)(51 122)(53 124)(55 126)(57 176)(58 101)(59 178)(60 103)(61 180)(62 105)(63 182)(64 107)(65 184)(66 109)(67 186)(68 111)(69 188)(70 85)(71 190)(72 87)(73 192)(74 89)(75 194)(76 91)(77 196)(78 93)(79 170)(80 95)(81 172)(82 97)(83 174)(84 99)(86 156)(88 158)(90 160)(92 162)(94 164)(96 166)(98 168)(100 142)(102 144)(104 146)(106 148)(108 150)(110 152)(112 154)(113 127)(115 129)(117 131)(119 133)(121 135)(123 137)(125 139)(141 175)(143 177)(145 179)(147 181)(149 183)(151 185)(153 187)(155 189)(157 191)(159 193)(161 195)(163 169)(165 171)(167 173)(198 212)(200 214)(202 216)(204 218)(206 220)(208 222)(210 224)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 29)(27 30)(28 31)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 85)(113 200)(114 201)(115 202)(116 203)(117 204)(118 205)(119 206)(120 207)(121 208)(122 209)(123 210)(124 211)(125 212)(126 213)(127 214)(128 215)(129 216)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 224)(138 197)(139 198)(140 199)(141 189)(142 190)(143 191)(144 192)(145 193)(146 194)(147 195)(148 196)(149 169)(150 170)(151 171)(152 172)(153 173)(154 174)(155 175)(156 176)(157 177)(158 178)(159 179)(160 180)(161 181)(162 182)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)
(1 204)(2 205)(3 206)(4 207)(5 208)(6 209)(7 210)(8 211)(9 212)(10 213)(11 214)(12 215)(13 216)(14 217)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 113)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 141)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 169)(94 170)(95 171)(96 172)(97 173)(98 174)(99 175)(100 176)(101 177)(102 178)(103 179)(104 180)(105 181)(106 182)(107 183)(108 184)(109 185)(110 186)(111 187)(112 188)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 162 204 77)(2 161 205 76)(3 160 206 75)(4 159 207 74)(5 158 208 73)(6 157 209 72)(7 156 210 71)(8 155 211 70)(9 154 212 69)(10 153 213 68)(11 152 214 67)(12 151 215 66)(13 150 216 65)(14 149 217 64)(15 148 218 63)(16 147 219 62)(17 146 220 61)(18 145 221 60)(19 144 222 59)(20 143 223 58)(21 142 224 57)(22 141 197 84)(23 168 198 83)(24 167 199 82)(25 166 200 81)(26 165 201 80)(27 164 202 79)(28 163 203 78)(29 185 114 109)(30 184 115 108)(31 183 116 107)(32 182 117 106)(33 181 118 105)(34 180 119 104)(35 179 120 103)(36 178 121 102)(37 177 122 101)(38 176 123 100)(39 175 124 99)(40 174 125 98)(41 173 126 97)(42 172 127 96)(43 171 128 95)(44 170 129 94)(45 169 130 93)(46 196 131 92)(47 195 132 91)(48 194 133 90)(49 193 134 89)(50 192 135 88)(51 191 136 87)(52 190 137 86)(53 189 138 85)(54 188 139 112)(55 187 140 111)(56 186 113 110)
G:=sub<Sym(224)| (1,15)(2,219)(3,17)(4,221)(5,19)(6,223)(7,21)(8,197)(9,23)(10,199)(11,25)(12,201)(13,27)(14,203)(16,205)(18,207)(20,209)(22,211)(24,213)(26,215)(28,217)(29,128)(30,44)(31,130)(32,46)(33,132)(34,48)(35,134)(36,50)(37,136)(38,52)(39,138)(40,54)(41,140)(42,56)(43,114)(45,116)(47,118)(49,120)(51,122)(53,124)(55,126)(57,176)(58,101)(59,178)(60,103)(61,180)(62,105)(63,182)(64,107)(65,184)(66,109)(67,186)(68,111)(69,188)(70,85)(71,190)(72,87)(73,192)(74,89)(75,194)(76,91)(77,196)(78,93)(79,170)(80,95)(81,172)(82,97)(83,174)(84,99)(86,156)(88,158)(90,160)(92,162)(94,164)(96,166)(98,168)(100,142)(102,144)(104,146)(106,148)(108,150)(110,152)(112,154)(113,127)(115,129)(117,131)(119,133)(121,135)(123,137)(125,139)(141,175)(143,177)(145,179)(147,181)(149,183)(151,185)(153,187)(155,189)(157,191)(159,193)(161,195)(163,169)(165,171)(167,173)(198,212)(200,214)(202,216)(204,218)(206,220)(208,222)(210,224), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,29)(27,30)(28,31)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,85)(113,200)(114,201)(115,202)(116,203)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,197)(139,198)(140,199)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194)(147,195)(148,196)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176)(157,177)(158,178)(159,179)(160,180)(161,181)(162,182)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188), (1,204)(2,205)(3,206)(4,207)(5,208)(6,209)(7,210)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,113)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,141)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,169)(94,170)(95,171)(96,172)(97,173)(98,174)(99,175)(100,176)(101,177)(102,178)(103,179)(104,180)(105,181)(106,182)(107,183)(108,184)(109,185)(110,186)(111,187)(112,188), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,162,204,77)(2,161,205,76)(3,160,206,75)(4,159,207,74)(5,158,208,73)(6,157,209,72)(7,156,210,71)(8,155,211,70)(9,154,212,69)(10,153,213,68)(11,152,214,67)(12,151,215,66)(13,150,216,65)(14,149,217,64)(15,148,218,63)(16,147,219,62)(17,146,220,61)(18,145,221,60)(19,144,222,59)(20,143,223,58)(21,142,224,57)(22,141,197,84)(23,168,198,83)(24,167,199,82)(25,166,200,81)(26,165,201,80)(27,164,202,79)(28,163,203,78)(29,185,114,109)(30,184,115,108)(31,183,116,107)(32,182,117,106)(33,181,118,105)(34,180,119,104)(35,179,120,103)(36,178,121,102)(37,177,122,101)(38,176,123,100)(39,175,124,99)(40,174,125,98)(41,173,126,97)(42,172,127,96)(43,171,128,95)(44,170,129,94)(45,169,130,93)(46,196,131,92)(47,195,132,91)(48,194,133,90)(49,193,134,89)(50,192,135,88)(51,191,136,87)(52,190,137,86)(53,189,138,85)(54,188,139,112)(55,187,140,111)(56,186,113,110)>;
G:=Group( (1,15)(2,219)(3,17)(4,221)(5,19)(6,223)(7,21)(8,197)(9,23)(10,199)(11,25)(12,201)(13,27)(14,203)(16,205)(18,207)(20,209)(22,211)(24,213)(26,215)(28,217)(29,128)(30,44)(31,130)(32,46)(33,132)(34,48)(35,134)(36,50)(37,136)(38,52)(39,138)(40,54)(41,140)(42,56)(43,114)(45,116)(47,118)(49,120)(51,122)(53,124)(55,126)(57,176)(58,101)(59,178)(60,103)(61,180)(62,105)(63,182)(64,107)(65,184)(66,109)(67,186)(68,111)(69,188)(70,85)(71,190)(72,87)(73,192)(74,89)(75,194)(76,91)(77,196)(78,93)(79,170)(80,95)(81,172)(82,97)(83,174)(84,99)(86,156)(88,158)(90,160)(92,162)(94,164)(96,166)(98,168)(100,142)(102,144)(104,146)(106,148)(108,150)(110,152)(112,154)(113,127)(115,129)(117,131)(119,133)(121,135)(123,137)(125,139)(141,175)(143,177)(145,179)(147,181)(149,183)(151,185)(153,187)(155,189)(157,191)(159,193)(161,195)(163,169)(165,171)(167,173)(198,212)(200,214)(202,216)(204,218)(206,220)(208,222)(210,224), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,29)(27,30)(28,31)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,85)(113,200)(114,201)(115,202)(116,203)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,197)(139,198)(140,199)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194)(147,195)(148,196)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176)(157,177)(158,178)(159,179)(160,180)(161,181)(162,182)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188), (1,204)(2,205)(3,206)(4,207)(5,208)(6,209)(7,210)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,121)(37,122)(38,123)(39,124)(40,125)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,113)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,141)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,169)(94,170)(95,171)(96,172)(97,173)(98,174)(99,175)(100,176)(101,177)(102,178)(103,179)(104,180)(105,181)(106,182)(107,183)(108,184)(109,185)(110,186)(111,187)(112,188), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,162,204,77)(2,161,205,76)(3,160,206,75)(4,159,207,74)(5,158,208,73)(6,157,209,72)(7,156,210,71)(8,155,211,70)(9,154,212,69)(10,153,213,68)(11,152,214,67)(12,151,215,66)(13,150,216,65)(14,149,217,64)(15,148,218,63)(16,147,219,62)(17,146,220,61)(18,145,221,60)(19,144,222,59)(20,143,223,58)(21,142,224,57)(22,141,197,84)(23,168,198,83)(24,167,199,82)(25,166,200,81)(26,165,201,80)(27,164,202,79)(28,163,203,78)(29,185,114,109)(30,184,115,108)(31,183,116,107)(32,182,117,106)(33,181,118,105)(34,180,119,104)(35,179,120,103)(36,178,121,102)(37,177,122,101)(38,176,123,100)(39,175,124,99)(40,174,125,98)(41,173,126,97)(42,172,127,96)(43,171,128,95)(44,170,129,94)(45,169,130,93)(46,196,131,92)(47,195,132,91)(48,194,133,90)(49,193,134,89)(50,192,135,88)(51,191,136,87)(52,190,137,86)(53,189,138,85)(54,188,139,112)(55,187,140,111)(56,186,113,110) );
G=PermutationGroup([(1,15),(2,219),(3,17),(4,221),(5,19),(6,223),(7,21),(8,197),(9,23),(10,199),(11,25),(12,201),(13,27),(14,203),(16,205),(18,207),(20,209),(22,211),(24,213),(26,215),(28,217),(29,128),(30,44),(31,130),(32,46),(33,132),(34,48),(35,134),(36,50),(37,136),(38,52),(39,138),(40,54),(41,140),(42,56),(43,114),(45,116),(47,118),(49,120),(51,122),(53,124),(55,126),(57,176),(58,101),(59,178),(60,103),(61,180),(62,105),(63,182),(64,107),(65,184),(66,109),(67,186),(68,111),(69,188),(70,85),(71,190),(72,87),(73,192),(74,89),(75,194),(76,91),(77,196),(78,93),(79,170),(80,95),(81,172),(82,97),(83,174),(84,99),(86,156),(88,158),(90,160),(92,162),(94,164),(96,166),(98,168),(100,142),(102,144),(104,146),(106,148),(108,150),(110,152),(112,154),(113,127),(115,129),(117,131),(119,133),(121,135),(123,137),(125,139),(141,175),(143,177),(145,179),(147,181),(149,183),(151,185),(153,187),(155,189),(157,191),(159,193),(161,195),(163,169),(165,171),(167,173),(198,212),(200,214),(202,216),(204,218),(206,220),(208,222),(210,224)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,29),(27,30),(28,31),(57,86),(58,87),(59,88),(60,89),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,99),(71,100),(72,101),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,109),(81,110),(82,111),(83,112),(84,85),(113,200),(114,201),(115,202),(116,203),(117,204),(118,205),(119,206),(120,207),(121,208),(122,209),(123,210),(124,211),(125,212),(126,213),(127,214),(128,215),(129,216),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,224),(138,197),(139,198),(140,199),(141,189),(142,190),(143,191),(144,192),(145,193),(146,194),(147,195),(148,196),(149,169),(150,170),(151,171),(152,172),(153,173),(154,174),(155,175),(156,176),(157,177),(158,178),(159,179),(160,180),(161,181),(162,182),(163,183),(164,184),(165,185),(166,186),(167,187),(168,188)], [(1,204),(2,205),(3,206),(4,207),(5,208),(6,209),(7,210),(8,211),(9,212),(10,213),(11,214),(12,215),(13,216),(14,217),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(22,197),(23,198),(24,199),(25,200),(26,201),(27,202),(28,203),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,121),(37,122),(38,123),(39,124),(40,125),(41,126),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,135),(51,136),(52,137),(53,138),(54,139),(55,140),(56,113),(57,142),(58,143),(59,144),(60,145),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,141),(85,189),(86,190),(87,191),(88,192),(89,193),(90,194),(91,195),(92,196),(93,169),(94,170),(95,171),(96,172),(97,173),(98,174),(99,175),(100,176),(101,177),(102,178),(103,179),(104,180),(105,181),(106,182),(107,183),(108,184),(109,185),(110,186),(111,187),(112,188)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,162,204,77),(2,161,205,76),(3,160,206,75),(4,159,207,74),(5,158,208,73),(6,157,209,72),(7,156,210,71),(8,155,211,70),(9,154,212,69),(10,153,213,68),(11,152,214,67),(12,151,215,66),(13,150,216,65),(14,149,217,64),(15,148,218,63),(16,147,219,62),(17,146,220,61),(18,145,221,60),(19,144,222,59),(20,143,223,58),(21,142,224,57),(22,141,197,84),(23,168,198,83),(24,167,199,82),(25,166,200,81),(26,165,201,80),(27,164,202,79),(28,163,203,78),(29,185,114,109),(30,184,115,108),(31,183,116,107),(32,182,117,106),(33,181,118,105),(34,180,119,104),(35,179,120,103),(36,178,121,102),(37,177,122,101),(38,176,123,100),(39,175,124,99),(40,174,125,98),(41,173,126,97),(42,172,127,96),(43,171,128,95),(44,170,129,94),(45,169,130,93),(46,196,131,92),(47,195,132,91),(48,194,133,90),(49,193,134,89),(50,192,135,88),(51,191,136,87),(52,190,137,86),(53,189,138,85),(54,188,139,112),(55,187,140,111),(56,186,113,110)])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D28 | C4○D28 | D4⋊2D7 |
| kernel | C23.14D28 | C14.C42 | C2×C4⋊Dic7 | C2×C23.D7 | C14×C22⋊C4 | C2×C28 | C22×C14 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 3 | 10 | 6 | 3 | 12 | 12 | 12 | 12 |
Matrix representation of C23.14D28 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 19 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,19,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[3,0,0,0,0,0,0,10,0,0,0,0,0,0,7,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,6,28],[0,3,0,0,0,0,10,0,0,0,0,0,0,0,0,7,0,0,0,0,25,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
C23.14D28 in GAP, Magma, Sage, TeX
C_2^3._{14}D_{28} % in TeX
G:=Group("C2^3.14D28"); // GroupNames label
G:=SmallGroup(448,487);
// by ID
G=gap.SmallGroup(448,487);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,387,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=c,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations